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Section 1-10 : Curvature

In this section we want to briefly discuss the curvature of a smooth curve (recall that for a smooth curve we require \(\vec r'\left( t \right)\) is continuous and \(\vec r'\left( t \right) \ne 0\)). The curvature measures how fast a curve is changing direction at a given point.

There are several formulas for determining the curvature for a curve. The formal definition of curvature is,

\[\kappa = \left| {\frac{{d\,\vec T}}{{ds}}} \right|\]

where \(\vec T\) is the unit tangent and \(s\) is the arc length. Recall that we saw in a previous section how to reparametrize a curve to get it into terms of the arc length.

In general the formal definition of the curvature is not easy to use so there are two alternate formulas that we can use. Here they are.